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At the end of the nineteenth century Lyapunov and Poincaré developed the so called qualitative theory of differential equations and introduced geometric- topological considerations which have led to the concept of dynamical systems. In its present abstract form this concept goes back to G.D. Birkhoff. This is also the starting point of Chapter 1 of this book in which uncontrolled and controlled time-continuous and time-discrete systems are investigated. Controlled dynamical systems could be considered as dynamical systems in the strong sense, if the controls were incorporated into the state space. We, however, adapt the conventional treatment of controlled systems as in control theory. We are mainly interested in the question of controllability of dynamical systems into equilibrium states. In the non-autonomous time-discrete case we also consider the problem of stabilization. We conclude with chaotic behavior of autonomous time discrete systems and actual real-world applications.



1. Uncontrolled Systems

The concept of dynamical systems has developed out of the qualitative theory of differential equations which was established by Lyapunov and Poincaré in the course of the two last decades of the nineteenth century. As final result of a development which lasted more than half a century the following abstract definition of a dynamical system has grown out:
Let X be a metric space with metric d. Further let I be an additive semigroup of real numbers, i.e. a subset I of \(\mathbb{R}\) with
$$\begin{array} {rlr} 0 \in I, & {}\\ t,s \in I & \Rightarrow t + s = s + t \in I,\\ {t, s, r} \in { I} & \Rightarrow {({t}{ + } s)}{ + }{ r}{ = }{ t + (s + r)} .\\ \end{array} $$
Werner Krabs, Stefan Pickl

2. Controlled Systems

We start with a system of differential equations of the form
$$ \dot{x}_i = f_i (x,u), \qquad i = 1,\ldots,n $$
where \( x \in {\mathbb R}^n, \,\, u \in {\mathbb R}^m, \)
$$ f_i : {\mathbb R}^n \times {\mathbb R}^m \to {\mathbb R} $$
with \( f_i \in C \left( {{\mathbb R}^{n + m}, {\mathbb R}} \right) \) and \( f_i \left( {\cdot, u} \right) \in C^1 \left( {\mathbb{R}^n, \mathbb{R}} \right) \) for every \( u\, \in \,\mathbb{R}^m \) and for i = 1, …, n.
Werner Krabs, Stefan Pickl

3. Chaotic Behavior of Autonomous Time-Discrete Systems

Let \( f : X \rightarrow X \) be a continuous mapping of a metric space X into itself. Then by the definition
$$ \pi \left( {x, n} \right) = f^n (x) \quad {\rm for\, all} \quad {x} \in {X} \quad {\rm and} \quad n \in {\mathbb N}_0 $$
$$ f^0 (x) = x \quad {\rm and} \quad f^{n + 1} (x) = f(f^n (x)), \quad n \in {\mathbb N}_0 ,$$
we obtain an autonomous time-discrete dynamical system (see Section 1.3.1).
Werner Krabs, Stefan Pickl

4. A Dynamical Method for the Calculation of Nash-Equilibria in n–Person Games

The fixed point theorem of Kakutani is a generalization of Bouwer’s fixed point theorem which says that every continuous mapping of a convex and compact subset of an n–dimensional Euclidean space into itself has at least one fixed point.
Werner Krabs, Stefan Pickl

5. Optimal Control in Chemotherapy of Cancer

We describe the time dependent size of the tumor by a real valued function \( T = T (t), \,\, t \in {\mathbb R}\) (t denotes the time), which we assume to be differentiable. The temporal development of this size (without treatment) we assume to be governed by the differential equation
$$ \dot{T}(t)= f(T(t))T(t), \quad t \in {\mathbb R}, $$
where the function \( f : {\mathbb R}_+ \rightarrow {\mathbb R}_+ \) is the growth rate of the tumor which is assumed to be in \(C^1 ({\mathbb R}_+)\) and to satisfy the condition
$$ f\prime(T) < 0 \quad {\rm for \, all} \quad T \geq 0. $$
Werner Krabs, Stefan Pickl


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